System and method for image reconstruction for helical cone beam computed tomography with factorized redundancy weighting

ABSTRACT

A method and system for computed tomography were weighted data is used to reconstruct an image. The apparatus includes an x-ray source producing a cone-beam of x-rays, an x-ray detector disposed to receive x-rays from the x-ray source, a data collection unit, and a processing unit for processing the data using a weighting function using contributions from fan-beam and cone-beam weighting functions. The apparatus can produce images with reduced artifacts.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to computed tomographic (CT) imaging, andin particular to weighting in helical cone-beam CT.

2. Discussion of the Background

As medical CT manufacturers produce scanners with increasing number ofdetector rows, there arises a need for a practical reconstructionalgorithm that can handle the increasing cone angle. Recently an exacthelical cone beam algorithm of the shift invariant FBP type (Katsevichalgorithm) was proposed, suitable for 1-PI and 3-PI reconstruction.After that practical ways to implement Katsevich algorithm in medical CTscanners were investigated. Generally, exact helical algorithms use onlydata within the helical PI-intervals, or, equivalently, within the N-PIwindow [10-11], where N=1, 3, . . . , is the number of helicalhalf-turns used in reconstruction. However from the practical point ofview N-PI window weighting has the following properties:

1) Some measured data located outside the N-PI window is not used, whichmeans extra dose to the patient.

2) All data within the N-PI window is used with the same weight; whileit is beneficial to the noise reduction, it makes an algorithm moresensitive to patient motion and

imperfections of real data.

3) The N-PI reconstruction restricts the choice of the helical pitch.For example, pitches in the range of 0.75-0.85 are too fast to be usedwith the 3-PI window, and are suboptimal to use with the 1-PI window,since only a small fraction of data is utilized.

On the other hand, 2D fan beam redundancy weighting has the followingadvantages:

1) Easily adjusted to the helical pitch

2) Smooth transition from 0 to 1 makes an algorithm more stable topatient motion and imperfections of real data.

SUMMARY OF THE INVENTION

The present invention is directed to a CT method and apparatus where, inone embodiment, the apparatus includes an x-ray source, an x-raydetector disposed to receive x-rays from the x-ray source, a datacollection unit, a processing unit for processing the data using aweighting function given as

${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$

where β is a projection angle of said x-rays, γ is a fan angle of saidx-rays, and ν is a detector coordinate parallel to axis of rotation ofsaid x-ray source.

In another embodiment, the method includes exposing a subject to x-raysfrom an x-ray source, collecting data, weighting the data using

${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$and reconstructing an image of the subject using the weighted data

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a diagram of a system according to the invention;

FIG. 2 is a diagram of scanning geometry;

FIG. 3 illustrates the fan-beam and cone-beam weighting functions;

FIG. 4 is a conventional image of a disk phantom;

FIG. 5 is an image of the disk phantom according to the invention;

FIG. 6 is a conventional image of a chest phantom;

FIG. 7 is an image of the chest phantom according to the invention; and

FIG. 8 is a diagram of conventional filtering and filtering according tothe invention.

DETAILED DESCRIPTION

FIG. 1 shows an x-ray computed topographic imaging device according tothe present invention. The projection data measurement systemconstituted by gantry 1 accommodates an x-ray source 3 that generates acone-beam of x-ray flux approximately cone-shaped, and a two-dimensionalarray type x-ray detector 5 consisting of a plurality of detectorelements arranged in two-dimensional fashion, i.e., a plurality ofelements arranged in one dimension stacked in a plurality of rows. X-raysource 3 and two-dimensional array type x-ray detector 5 are installedon a rotating ring 2 in facing opposite sides of a subject, who is laidon a sliding sheet of a bed 6. Two-dimensional array type x-ray detector5 is mounted on rotating ring 2. Each detector element will correspondwith one channel. X-rays from x-ray source 3 are directed on to subjectthrough an x-ray filter 4. X-rays that have passed through the subjectare detected as an electrical signal by two-dimensional array type x-raydetector 5.

X-ray controller 8 supplies a trigger signal to high voltage generator7. High voltage generator 7 applies high voltage to x-ray source 3 withthe timing with which the trigger signal is received. This causes x-raysto be emitted from x-ray source 3. Gantry/bed controller 9 synchronouslycontrols the revolution of rotating ring 2 of gantry 1 and the slidingof the sliding sheet of bed 6. System controller 10 constitutes thecontrol center of the entire system and controls x-ray controller 8 andgantry/bed controller 9 and x-rays are emitted continuously orintermittently at fixed angular intervals from x-ray source 3.

The output signal of two-dimensional array type x-ray detector 5 isamplified by a data collection unit 11 for each channel and converted toa digital signal, to produce projection data. The projection data thatis output from data collection unit 11 is fed to processing unit 12.Processing unit 12 performs various processing using the projectiondata. Unit 12 performs data sampling and shifting (described in moredetail below), filtering, backprojection and reconstruction, as well asother desired operation on the projection data. Unit 12 determinesbackprojection data reflecting the x-ray absorption in each voxel. Inthe circular scanning system using a cone-beam of x-rays as in the firstembodiment, the imaging region (effective field of view) is ofcylindrical shape of radius R centered on the axis of revolution. Unit12 defines a plurality of voxels (three-dimensional pixels) in thisimaging region, and finds the backprojection data for each voxel. Thethree-dimensional image data or tomographic image data compiled by usingthis backprojection data is sent to display device 14, where it isdisplayed visually as a three-dimensional image or tomographic image.

The system geometry for explaining the apparatus and method according tothe invention is shown in FIG. 2. The apparatus and system are directedto a cone-beam system. X-rays are emitted at a plurality of projectionangles around the object to collect at least minimally requiredtomographic data. At each projection angle β, x-rays are emitted with acone angle γ and a fan angle α and pass through an object with a scannedfield of view. The source follows a helical trajectory. The rays aredetected by the two-dimensional detector.

A two-dimensional detector can be described where each detector elementk, k=1 . . . Nseg×M, where Nseg is the number of detector rows, and M isthe number of elements per detector row.

Intensity of the x-ray photon beam (ray) at the detector element k,attenuated by an object or patient, is given by:I _(k) =I _(k) ⁰ exp(−∫μ(x)dx),  (1)where μ(x) is the attenuation function to be reconstructed, I_(k) ⁰ isthe beam intensity before attenuation by μ(x), as produced by the x-raytube and after penetrating through the x-ray (wedge, bowtie) filter, and∫μ(x)dx is the line integral of μ(x) along the line l. Mathematically,μ(x) can be reconstructed given a set of line integrals corresponding toa plurality of lines l. Therefore, measured intensity data are to beconverted into line integrals first∫μ(x)dx=ln(I _(k) ⁰)−ln(I _(k))  (2)

X-ray tomographic reconstruction consists of the following three mainsteps, data acquisition, data processing and data reconstruction. Indata acquisition, the x-ray intensity data are collected at eachdetector element and each predefined angular view position. This is donewithin rotating part of the gantry 1. Detector 5 measures incident x-rayflux and converts it into an electric signal. There are two main typesof detectors: energy (charge) integrating and photon counting. Theelectric signal is transferred from a rotating part of the gantry 1 tostationary part though the slipring 2. During this step data may becompressed.

In data processing, data is converted from x-ray intensity measurementsto the signal corresponding to line integrals according to equation (2).Also, various corrections steps are applied to reduce effects ofundesired physical phenomena, such as scatter, x-ray beam hardening,compensate non-uniform response function of each detector element, andto reduce noise.

Depending on the algorithm, data reconstruction contains all or some ofthe following processing steps:

-   -   Cosine (fan angle, cone angle) weighting (can be ×cos, or 1/cos)    -   Data differentiation. It can be performed with respect to fan        angle, cone angle, projection angle, source trajectory        coordinate, vertical detector coordinate, horizontal detector        coordinate, or any combination of those.    -   Data redundancy weighting. Data is multiplied by the weight        function W, which may be a function of fan angle, cone angle,        projection angle, source trajectory coordinate, vertical        detector coordinate, horizontal detector coordinate, or any        combination of those.    -   Convolution (filtering). This step utilizes convolution kernel.        Some algorithms use ramp-based kernel (H(w)=|w|), some use        Hilbert-based kernel (h(t)=1/t, h(t)=1/sin(t), H(w)=i sign(w)).        Kernels can be adjusted to the fan beam geometry, scaled,        modulated, apodised, modified, or any combination of those.        Filtering can be applied along non-horizontal directions, in        which case data resampling to the filtering directions and        possibly inverse resampling to the original detector grid is        required.    -   Backprojection. In this step data is projected back in the image        domain. Usually, backprojected data is weighted by a distance        factor. Distance factor is inversely proportional to the        distance L from the x-ray source position to the reconstructed        pixel. Distance factor can be proportional to 1/L or 1/L². Also,        some additional data redundancy weighting can be applied here on        the pixel-by-pixel basis. Also, usually backprojection step        includes obtaining data value corresponding to the ray through        the reconstructed pixel by either data interpolation or data        extrapolation. This process can be done in a numerous variety of        ways.        The order in which the above steps are applied depends on a        specific reconstruction algorithm.

In the present invention, the processing unit performs data redundancyweighting, termed smooth cone beam weighting, in the following manner.Here, β is the projection angle, γ is the fan angle, ν is the detectorcoordinate parallel to the axis of rotation, α is the cone angle whereν=Rtan α, g (β, γ, ν) is the cone beam data along the helical sourcetrajectory λ(β)=(R cos β, R sin β, βH/2π), with radius R and pitch H.The physical size of the detector limits detector coordinates to−ν_(max)<ν<ν_(max), and −γ_(max)<γ<γ_(max). In reconstructing an imageplane P, normally horizontal planes are reconstructed, so P is given bythe equation z=z₀. The values of ν are within the scan range [β_(start),β_(end)], which depends on P.

For each data sample (view, ch, seg), and corresponding ray (β, γ, ν)the following steps are performed:

locate a corresponding image pixel x, which is found by an intersectionof the reconstruction plane P and ray (β, γ, ν),

find all measured complementary rays (β_(n), γ_(n), ν_(n)), where n=N .. . −N, through x and N is the number of helical half turns. Thecomplementary coordinates are given by:

${\beta_{n}\left( {\beta,\gamma} \right)} = \left\{ {{\begin{matrix}{{\beta + {2\gamma} + {n\;\pi}},{n\mspace{14mu}{odd}}} \\{{\beta + {2\; n\;\pi}},{n\mspace{14mu}{even}},}\end{matrix}{\gamma_{n}(\gamma)}} = \left\{ {{\begin{matrix}{{- \gamma},{n\mspace{14mu}{odd}}} \\{\gamma,{n\mspace{14mu}{even}},}\end{matrix}{v_{n}\left( {\beta,\gamma,v} \right)}} = \left\{ \begin{matrix}{{\Delta\; z_{n}{R/L^{C}}},{n\mspace{14mu}{odd}}} \\{{\Delta\; z_{n}{R/L}},{n\mspace{14mu}{even}}}\end{matrix} \right.} \right.} \right.$

where Δz_(n)=Δβ_(n)H/2; and ρβ_(n)=β_(S)−β_(n), where β_(S) is the viewangle corresponding to the image slice position (z₀=β_(S)H/2;), H is thehelical pitch, L=Δz R/ν and L^(c)=2 R cos γ−L, and

weight the data g(β, γ, ν) depending on the ray position, and normalizeby the weighted contributions of all complementary rays.

The cone beam weighting function is given by:

${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$Note that in the summation the index n=0 corresponds to the direct ray:β₀=β, and ν₀=ν.

Examples of functions u_(FB)(β) (fan beam) and u_(CB)(ν, γ) (cone beam)are shown in FIG. 3. The functions u_(FB)(β) and u_(CB)(ν, γ) are calledauxiliary weighting functions. Here Δβ is called the smoothing interval.It can be given as a fixed number of views, as a fixed angular range(for example 10°), or as a percentage of the angular scan length(β_(end)−β_(start)). It can be as small as 0% or as large as 50%.

The function u_(FB)(β) is given by:

${u_{FB}(\beta)} = \left\{ \begin{matrix}{0,} & {\beta < \beta_{start}} \\{{p\left( \frac{\beta - \beta_{start}}{\Delta\beta} \right)},} & {\beta_{start} \leq \beta < {\beta_{start} + {\Delta\beta}}} \\{1,} & {{\beta_{start} + {\Delta\beta}} \leq \beta \leq {\beta_{end} - {\Delta\beta}}} \\{{p\left( \frac{\beta_{end} - \beta}{\Delta\beta} \right)},} & {{\beta_{end} - {\Delta\beta}} < \beta \leq \beta_{end}} \\{0,} & {\beta > \beta_{end}}\end{matrix} \right.$

Here the function p( ) can be chosen in various ways. Some examples:

$\begin{matrix}{{Linear}\text{:}} & {{p(x)} = {x.}} \\{{Polynominal}\text{:}} & {{p(x)} = {{3x^{2}} - {2{x^{3}.}}}} \\{{Trigonometric}\text{:}} & {{p(x)} = {{\frac{1}{2}\left( {1 - {\cos\left( {\pi\; x} \right)}} \right)\mspace{14mu}{or}\mspace{14mu}{p(x)}} = {\sin^{2}\left( \frac{\pi\; x}{2} \right)}}}\end{matrix}$In general, function p( ) is any function that satisfies: p(0)=0,p(1)=1, and p monotonically increases from 0 to 1.

The function u_(CB)(ν, γ) can be implemented in various ways. Forexample it can be given by:

${u_{CB}(v)} = \left\{ \begin{matrix}{0,} & {v < {- v_{\max}}} \\{{p\left( \frac{v + v_{\max}}{\Delta\; v} \right)},} & {{- v_{\max}} \leq v < {{- v_{\max}} + {\Delta\; v}}} \\{1,} & {{{- v_{\max}} + {\Delta\; v}} \leq v \leq {v_{\max} - {\Delta\; v}}} \\{p\left( \frac{v_{\max} - v}{\Delta\; v} \right)} & {{v_{\max} - {\Delta\; v}} < v \leq v_{\max}} \\{0,} & {v > v_{\max}}\end{matrix} \right.$Here Δν is called the smoothing interval. It can be given as a fixedlength (for example 3.2 mm, or 3.2 segments), or as a percentage of thedetector height 2ν_(max). It can be as small as 0% or as large as 50%.The function p( ) can be chosen as discussed above.

Another version, u_(CB)(ν, γ) can be given by:

${u_{CB}\left( {v,\gamma} \right)} = \left\{ \begin{matrix}{0,} & {v < {{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}}} \\{{p\left( \frac{v - {v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}}{2\delta\;{v_{-}(\gamma)}} \right)},} & {{{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}} \leq v < {{- {v_{-}(\gamma)}} + {\delta\;{v_{-}(\gamma)}}}} \\{1,} & {{{v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}} \leq v \leq {{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}}} \\{{p\left( \frac{{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}} - v}{2\delta\;{v_{+}(\gamma)}} \right)},} & {{{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}} < v \leq {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}} \\{0,} & {v > {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}}\end{matrix} \right.$where ν₊(γ) and ν⁻(γ) are the boundaries of the n-PI window and aregiven by:

${v_{+}\left( {\gamma,n} \right)} = \frac{H\left( {{2\gamma}\; + {n\;\pi}} \right)}{4\pi\;\cos\;\gamma}$${v_{-}\left( {\gamma,n} \right)} = \frac{H\left( {{2\gamma} - {n\;\pi}} \right)}{4\pi\;\cos\;\gamma}$and n is equal to either 1 or 3, depending on the helical pitch. Hereδν₊(γ) and δν⁻(γ) are called the smoothing intervals. They can be givenas a fixed length (for example 3.2 mm, or 3.2 segments), or as apercentage of the detector height 2ν_(max). It can be as small as 0% oras large as 50%. It can also be proportional to:δν₊(γ)=C(ν_(max)−ν₊(γ))δν⁻(γ)=C(ν_(max)+ν⁻(γ)),where C is the proportionality constant, 0<=C<=1.

The function p( ) can be chosen as discussed above, except for it shouldsatisfy an additional condition: p(0.5)=0.5.

Images prepared according to the invention are shown in FIGS. 4-7. FIG.4 shows a conventional image of a disk phantom obtained using a 320 by0.5 detector and a helical pitch of 136 mm/rev. The image shown in FIG.5 was obtained under the same conditions but was processed according tothe invention. FIG. 5 is of much better quality.

Similarly, FIG. 6 is a conventional image of a chest phantom while FIG.7 is an image obtained using the present invention. Again, FIG. 7 is ofmuch better quality where the appearance of many artifacts is reduced

One possible modification of the proposed method is when filtering isapplied along non-horizontal filtering directions, as shown in FIG. 8.

The present invention may be implemented in software or in hardware. Inparticular the operation of the processing unit described above can becarried out as a software program run on a microprocessor or a computer.The software can be stored on a computer-readable medium and loaded intothe system.

Numerous other modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

The invention claimed is:
 1. A computed-tomography apparatus,comprising: an x-ray source; an x-ray detector disposed to receivex-rays from said x-ray source; a data collection unit; a processing unitfor processing said data using a weighting function given as${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$where β is a projection angle of said x-rays, γ is a fan angle of saidx-rays, N is a number of helical half-turns, n is an index ofcomplementary rays and ν is a detector coordinate parallel to axis ofrotation of said x-ray source, and wherein said processing uses saidweighting function where:${u_{CB}\left( {v,\gamma} \right)} = \left\{ \begin{matrix}{0,} & {v < {{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}}} \\{{p\left( \frac{v - {v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}}{2\delta\;{v_{-}(\gamma)}} \right)},} & {{{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}} \leq v < {{- {v_{-}(\gamma)}} + {\delta\;{v_{-}(\gamma)}}}} \\{1,} & {{{v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}} \leq v \leq {{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}}} \\{{p\left( \frac{{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}} - v}{2\delta\;{v_{+}(\gamma)}} \right)},} & {{{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}} < v \leq {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}} \\{0,} & {v > {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}}\end{matrix} \right.$ where ν₊(γ) and ν⁻(γ) are boundaries of an n-PIwindow and δν₊(γ) and δν⁻(γ) are smoothing intervals.
 2. An apparatus asrecited in claim 1, wherein said processing uses said weighting functionwhere: ${u_{FB}(\beta)} = \left\{ \begin{matrix}{0,} & {\beta < \beta_{start}} \\{{p\left( \frac{\beta - \beta_{start}}{\Delta\beta} \right)},} & {\beta_{start} \leq \beta < {\beta_{start} + {\Delta\beta}}} \\{1,} & {{\beta_{start} + {\Delta\beta}} \leq \beta \leq {\beta_{end} - {\Delta\beta}}} \\{{p\left( \frac{\beta_{end} - \beta}{\Delta\beta} \right)},} & {{\beta_{end} - {\Delta\beta}} < \beta \leq \beta_{end}} \\{0,} & {\beta > \beta_{end}}\end{matrix} \right.$ where β_(start) and β_(end) are the start and endangles of a scan, respectively, p( ) is any function that satisfies:p(0)=0, p(1)=1, and p monotonically increases from 0 to
 1. 3. Anapparatus as recited in claim 1, wherein said processing unit comprisesa filtering sub-unit that applies filtering along detector rows.
 4. Anapparatus as recited in claim 1, wherein said processing unit comprisesa filtering sub-unit that applies filtering in directions other thanalong detector rows.
 5. An apparatus as recited in claim 1, wherein saidprocessing uses said weighting function where:βν₊(γ)=C(ν_(max)−ν₊(γ))βν⁻(γ)=C(ν_(max)+ν⁻(γ))v₊(γ, n)=H(2γ+nπ)/4πcosγv⁻(γ, n)=H(2γ−nπ)/4πcosγ C is a proportionality constant, 0<C<1, nequals 1 or 3, and H is a helical pitch.
 6. A computed tomographymethod, comprising: exposing a subject to x-rays from an x-ray source;collecting data; processing said data comprising weighting said datausing a weighting function given as${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$where β is a projection angle of said x-rays, γ is a fan angle of saidx-rays, N is a number of helical half turns, n is an index ofcomplementary rays and ν is a detector coordinate parallel to axis ofrotation of said x-ray source; and reconstructing an image of saidsubject using said weighted data, wherein said processing comprisesusing said weighting function where:${u_{CB}\left( {v,\gamma} \right)} = \left\{ \begin{matrix}{0,} & {v < {{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}}} \\{{p\left( \frac{v - {v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}}{2\delta\;{v_{-}(\gamma)}} \right)},} & {{{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}} \leq v < {{- {v_{-}(\gamma)}} + {\delta\;{v_{-}(\gamma)}}}} \\{1,} & {{{v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}} \leq v \leq {{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}}} \\{{p\left( \frac{{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}} - v}{2\delta\;{v_{+}(\gamma)}} \right)},} & {{{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}} < v \leq {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}} \\{0,} & {v > {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}}\end{matrix} \right.$ where ν₊(γ) and ν⁻(γ) are boundaries of an n-PIwindow, δν₊(γ) and δν⁻(γ) are smoothing intervals and p( )is anyfunction that satisfies: p(0)=0, p(0.5)=0.5, p(1)=1, and p monotonicallyincreases from 0 to
 1. 7. A method as recited in claim 6, wherein saidprocessing comprises using said weighting function where:${u_{FB}(\beta)} = \left\{ \begin{matrix}{0,} & {\beta < \beta_{start}} \\{{p\left( \frac{\beta - \beta_{start}}{\Delta\beta} \right)},} & {\beta_{start} \leq \beta < {\beta_{start} + {\Delta\beta}}} \\{1,} & {{\beta_{start} + {\Delta\beta}} \leq \beta \leq {\beta_{end} - {\Delta\beta}}} \\{{p\left( \frac{\beta_{end} - \beta}{\Delta\beta} \right)},} & {{\beta_{end} - {\Delta\beta}} < \beta \leq \beta_{end}} \\{0,} & {\beta > \beta_{end}}\end{matrix} \right.$ where p( )is any function that satisfies: p(0)=0,p(1)=1, β_(start) and β_(end) are the start and end angles of a scan,respectively, and p monotonically increases from 0 to
 1. 8. A method asrecited in claim 6, wherein said processing comprises filtering alongdetector rows.
 9. A method as recited in claim 6, wherein saidprocessing comprises filtering in directions other than along detectorrows.
 10. A method as recited in claim 6, wherein said processingcomprises using said weighting function where:βν₊(γ)=C(ν_(max)−ν₊(γ))βν⁻(γ)=C(ν_(max)+ν⁻(γ))v₊(γ, n)=H(2γ+nπ)/4πcosγv⁻(γ, n)=H(2γ−nπ)/4πcosγ C is a proportionality constant, 0<C<1, nequals 1 or 3, and H is a helical pitch.
 11. A non-transitorycomputer-readable medium containing instructions that may be executed bya computer to perform a method comprising: collecting data associatedwith exposing a subject to x-rays from an x-ray source; weighting saiddata using a weighting function given as${w_{CB}\left( {\beta,\gamma,v} \right)} = \frac{{u_{FB}(\beta)} \cdot {u_{CB}\left( {v,\gamma} \right)}}{\sum\limits_{n = {- N}}^{N}\left( {{u_{FB}\left( {\beta_{n}\left( {\beta,\gamma} \right)} \right)} \cdot {u_{CB}\left( v_{n} \right)}} \right)}$where β is a projection angle of said x-rays, γ is a fan angle of saidx-rays, N is a number of helical half turns, n is an index ofcomplementary rays and ν is a detector coordinate parallel to axis ofrotation of said x-ray source; and reconstructing an image of saidsubject using said weighted data, where:${u_{CB}\left( {v,\gamma} \right)} = \left\{ \begin{matrix}{0,} & {v < {{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}}} \\{{p\left( \frac{v - {v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}}{2\delta\;{v_{-}(\gamma)}} \right)},} & {{{v_{-}(\gamma)} - {\delta\;{v_{-}(\gamma)}}} \leq v < {{- {v_{-}(\gamma)}} + {\delta\;{v_{-}(\gamma)}}}} \\{1,} & {{{v_{-}(\gamma)} + {\delta\;{v_{-}(\gamma)}}} \leq v \leq {{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}}} \\{{p\left( \frac{{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}} - v}{2\delta\;{v_{+}(\gamma)}} \right)},} & {{{v_{+}(\gamma)} - {\delta\;{v_{+}(\gamma)}}} < v \leq {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}} \\{0,} & {v > {{v_{+}(\gamma)} + {\delta\;{v_{+}(\gamma)}}}}\end{matrix} \right.$ and where ν₊(γ) and ν⁻(γ) are boundaries of ann-PI window, βν₊(γ) and βν⁻(γ) are smoothing intervals and p( ) is anyfunction that satisfies: p(0)=0, p(0.5) =0.5, p(1)=1, and pmonotonically increases from 0 to
 1. 12. A medium as recited in claim11, wherein said method comprises using said weighting function where:${u_{FB}(\beta)} = \left\{ \begin{matrix}{0,} & {\beta < \beta_{start}} \\{{p\left( \frac{\beta - \beta_{start}}{\Delta\beta} \right)},} & {\beta_{start} \leq \beta < {\beta_{start} + {\Delta\beta}}} \\{1,} & {{\beta_{start} + {\Delta\beta}} \leq \beta \leq {\beta_{end} - {\Delta\beta}}} \\{{p\left( \frac{\beta_{end} - \beta}{\Delta\beta} \right)},} & {{\beta_{end} - {\Delta\beta}} < \beta \leq \beta_{end}} \\{0,} & {\beta > \beta_{end}}\end{matrix} \right.$ where p( ) is any function that satisfies: p(0)=0,p(1)=1, β_(start) and β_(end) are the start and end angles of a scan,respectively, and p monotonically increases from 0 to
 1. 13. A medium asrecited in claim 11, wherein said method comprises filtering alongdetector rows.
 14. A medium as recited in claim 11, wherein said methodcomprises filtering in directions other than along detector rows.
 15. Amedium as recited in claim 11, wherein said method comprises using saidweighting function where:βν₊(γ)=C(ν_(max)−ν₊(γ))βν⁻(γ)=C(ν_(max)+ν⁻(γ))v₊(γ, n)=H(2γ+nπ)/4πcosγv⁻(γ, n)=H(2γ−nπ)/4πcosγ C is a proportionality constant, 0<C<1, nequals 1 or 3, and H is a helical pitch.